Mid-Frequency Range Acoustic Radiation From Slender Elastic Bodies Using the Surface Variational Principle

Abstract

The surface variational principle (SVP) yields analytical-type results for radiation and scattering from submerged bodies whose shape does not suit classical techniques for analyzing the Helmholtz equation. The approach employs Ritz series expansions for surface pressure and velocity in the frequency domain. The relation between the series coefficients is obtained by extremizing the SVP functional. The present work extends the earlier developments to the case of an axisymmetric elastic shell that is subjected to an arbitrary excitation. The surface pressure and normal velocity are represented as a sequence of surface waves that are the trace of the waves in the surrounding fluid medium. SVP is used to determine the wavenumber spectrum of pressure amplitudes generated by a specific wave having unit velocity amplitude. The structural displacement field is also represented by Ritz expansions, and equations governing the generalized coordinates associated with these series are obtained by invoking Hamilton’s principle. Difficulties in satisfying the continuity conditions at the apexes are circumvented by selecting basis functions that map spherical shell eigenmodes onto the surface of the shell. The structural dynamic equations are coupled to the SVP equations by matching the normal velocity in the fluid to the time derivative of the normal displacement, as well as using the series expansion for surface pressure to form the acoustic contribution to the generalized forces. Results for a spherical shell subjected to a transverse point force at the equator, which is a nonaxisymmetric representation of the excitation, are compared with analytic results. Predictions for a long hemi-capped cylindrical shell in the mid-frequency range are compared to those obtained from SARA-2D (Allik, 1991), which is a finite/infinite element program. In addition to providing validation of the SVP implementation, the cylinder example is used to illustrate the convergence and error measures provided by an SVP analysis.